kinetic energy

Kinetic energy is the energy a body has by virtue of being in motion. For a point mass moving at speed v, it equals ½·m·v². The formula is not arbitrary: it drops out of integrating F = m·a along the path, and it is exactly the work that must be done to bring the body from rest to speed v.

The v² dependence is essential. It is why a car at 60 mph carries four times the kinetic energy of the same car at 30 mph, and why its stopping distance scales quadratically with speed — braking must dissipate four times as much energy. It is also why Leibniz's 1686 proposal of vis viva (m·v², missing a factor of two) was closer to the truth than Descartes's m·|v| — kinetic energy, not linear momentum, is the quantity that matches the work done to set a body in motion.

Kinetic energy combines with potential energy to form the total mechanical energy of a system. In the absence of friction or other dissipative forces the total is conserved — the pendulum's swing, the falling brick, the orbiting planet. When kinetic energy is apparently lost, as in a car crash or a falling meteor, it has been converted to heat, sound, or deformation — never destroyed, only moved elsewhere in the energy budget.